Topological data analysis (TDA) is an emerging field at the interface between algebraic topology and data science. The philosophy behind TDA is to leverage invariants from algebraic topology to gain insights into data sets. While initially, TDA was developed and promoted by mathematicians, it is now applied in a variety of disciplines such as biology, chemistry, and materials science. The key tool in TDA is the persistence diagram, which allows capturing the appearance and disappearance of topological features at multiple scales. In this lecture, I will review recent advances in the statistical foundations of dealing with persistence diagrams on random input.